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Given a metric space $\mathcal{M}=(X,\delta)$, a weighted graph $G$ over $X$ is a metric $t$-spanner of $\mathcal{M}$ if for every $u,v \in X$, $\delta(u,v)\le d_G(u,v)\le t\cdot \delta(u,v)$, where $d_G$ is the shortest path metric in $G$. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points $(s_1, \ldots, s_n)$, where the points are presented one at a time (i.e., after $i$ steps, we saw $S_i = \{s_1, \ldots , s_i\}$). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a $t$-spanner $G_i$ for $S_i$ for all $i$, while minimizing the number of edges, and their total weight. We construct online $(1+\varepsilon)$-spanners in Euclidean $d$-space, $(2k-1)(1+\varepsilon)$-spanners for general metrics, and $(2+\varepsilon)$-spanners for ultrametrics. Most notably, in Euclidean plane, we construct a $(1+\varepsilon)$-spanner with competitive ratio $O(\varepsilon^{-3/2}\log\varepsilon^{-1}\log n)$, bypassing the classic lower bound $\Omega(\varepsilon^{-2})$ for lightness, which compares the weight of the spanner, to that of the MST.

This paper provides a new version of matrix semi-tensor product method based on adjacent transpositions to test symmetric games. The advantage of using adjacent transpositions lies in the great simplification of analysis of symmetric games. By using the new method, new necessary and sufficient conditions for symmetric games are proposed, and a group of bases of a symmetric game space can be easily calculated. Moreover, the testing equations with the minimum number can be concretely determined. Finally, two examples are displayed to show the effectiveness of the proposed method.

We consider the well-studied problem of decomposing a vector time series signal into components with different characteristics, such as smooth, periodic, nonnegative, or sparse. We propose a simple and general framework in which the components are defined by loss functions (which include constraints), and the signal decomposition is carried out by minimizing the sum of losses of the components (subject to the constraints). When each loss function is the negative log-likelihood of a density for the signal component, our method coincides with maximum a posteriori probability (MAP) estimation; but it also includes many other interesting cases. We give two distributed optimization methods for computing the decomposition, which find the optimal decomposition when the component class loss functions are convex, and are good heuristics when they are not. Both methods require only the masked proximal operator of each of the component loss functions, a generalization of the well-known proximal operator that handles missing entries in its argument. Both methods are distributed, i.e., handle each component separately. We derive tractable methods for evaluating the masked proximal operators of some loss functions that, to our knowledge, have not appeared in the literature.

Center-based clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is undoubtedly the k-means problem, which, given a set $P$ of points from a metric space and a parameter $k<|P|$, requires to determine a subset $S$ of $k$ centers minimizing the sum of all squared distances of points in $P$ from their closest center. A more general formulation, known as k-means with $z$ outliers, introduced to deal with noisy datasets, features a further parameter $z$ and allows up to $z$ points of $P$ (outliers) to be disregarded when computing the aforementioned sum. We present a distributed coreset-based 3-round approximation algorithm for k-means with $z$ outliers for general metric spaces, using MapReduce as a computational model. Our distributed algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term $O(\gamma)$ away from the one achievable by the best known sequential (possibly bicriteria) algorithm, where $\gamma$ can be made arbitrarily small. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by the doubling dimension $D$ of the metric space. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for general metrics.

Model-based methods for recommender systems have been studied extensively for years. Modern recommender systems usually resort to 1) representation learning models which define user-item preference as the distance between their embedding representations, and 2) embedding-based Approximate Nearest Neighbor (ANN) search to tackle the efficiency problem introduced by large-scale corpus. While providing efficient retrieval, the embedding-based retrieval pattern also limits the model capacity since the form of user-item preference measure is restricted to the distance between their embedding representations. However, for other more precise user-item preference measures, e.g., preference scores directly derived from a deep neural network, they are computationally intractable because of the lack of an efficient retrieval method, and an exhaustive search for all user-item pairs is impractical. In this paper, we propose a novel method to extend ANN search to arbitrary matching functions, e.g., a deep neural network. Our main idea is to perform a greedy walk with a matching function in a similarity graph constructed from all items. To solve the problem that the similarity measures of graph construction and user-item matching function are heterogeneous, we propose a pluggable adversarial training task to ensure the graph search with arbitrary matching function can achieve fairly high precision. Experimental results in both open source and industry datasets demonstrate the effectiveness of our method. The proposed method has been fully deployed in the Taobao display advertising platform and brings a considerable advertising revenue increase. We also summarize our detailed experiences in deployment in this paper.

We introduce Monte-Carlo Attention (MCA), a randomized approximation method for reducing the computational cost of self-attention mechanisms in Transformer architectures. MCA exploits the fact that the importance of each token in an input sequence varies with respect to their attention scores; thus, some degree of error can be tolerable when encoding tokens with low attention. Using approximate matrix multiplication, MCA applies different error bounds to encode input tokens such that those with low attention scores are computed with relaxed precision, whereas errors of salient elements are minimized. MCA can operate in parallel with other attention optimization schemes and does not require model modification. We study the theoretical error bounds and demonstrate that MCA reduces attention complexity (in FLOPS) for various Transformer models by up to 11$\times$ in GLUE benchmarks without compromising model accuracy.

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.

For many computer vision applications such as image captioning, visual question answering, and person search, learning discriminative feature representations at both image and text level is an essential yet challenging problem. Its challenges originate from the large word variance in the text domain as well as the difficulty of accurately measuring the distance between the features of the two modalities. Most prior work focuses on the latter challenge, by introducing loss functions that help the network learn better feature representations but fail to account for the complexity of the textual input. With that in mind, we introduce TIMAM: a Text-Image Modality Adversarial Matching approach that learns modality-invariant feature representations using adversarial and cross-modal matching objectives. In addition, we demonstrate that BERT, a publicly-available language model that extracts word embeddings, can successfully be applied in the text-to-image matching domain. The proposed approach achieves state-of-the-art cross-modal matching performance on four widely-used publicly-available datasets resulting in absolute improvements ranging from 2% to 5% in terms of rank-1 accuracy.

We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $\epsilon$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/\epsilon)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/\epsilon^2)$ bound (SODA'03), and bypasses an $\Omega(d^2/\epsilon^2)$ lower bound of (KDD'14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetilde{\Theta}(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $\epsilon$. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix $A$ more generally, which includes the stable rank, Schatten-$p$ norms, and SVD entropy. Specifically, we propose a bounded entry model, where $A$ is required to have entries bounded by $1$ in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.